Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence (IJCAI-18) Finite Model Reasoning in Hybrid Classes of Existential Rules Georg Gottlob1, Marco Manna2 and Andreas Pieris3 1Department of Computer Science, University of Oxford 2Department of Mathematics and Computer Science, University of Calabria 3School of Informatics, University of Edinburgh [email protected], [email protected], [email protected] Abstract ness is well-suited for forward reasoning, which in turn relies on the fact that guarded rules admit tree-like universal mod- Two paradigmatic restrictions that have been stud- els. However, stickiness does not enjoy the above tree model ied for ensuring the decidability of query answering property, and instead it relies on backward reasoning. under existential rules are guardedness and sticki- As one might expect, there are useful statements that can ness. With the aim of consolidating these restric- be expressed via guarded rules, but not with sticky rules, and tions, a flexible condition, called tameness, has vice versa. For example, using guarded rules we can state that been proposed a few years ago, which relies on hy- the supervisor of a senior employee is also a senior employee, brid reasoning, i.e., a combination of forward and which is provably not expressible via sticky rules: backward procedures. The complexity of query an- swering under this hybrid class of existential rules SeniorEmp(x); HasSupervisor(x; y) ! SeniorEmp(y): is by now well-understood. However, the complex- ity of finite query answering, i.e., query answering This rule is guarded since it has an atom in its body, called a under finite models, has remained an open problem. guard, that contains x and y. However, using guarded rules Closing this problem is the main goal of this work. we cannot say, e.g., that senior employees earn more money than junior employees. This is expressible via the sticky rule: 1 Introduction SeniorEmp(x); JuniorEmp(y) ! MoreThan(x; y): Rule-based languages lie at the core of knowledge representa- The goal of stickiness is to capture joins that are not express- tion and databases. In knowledge representation they are used ible via guarded rules. This is done by forcing the join vari- for declarative problem solving, and, more recently, to model ables to be propagated to the inferred atoms. The above rule and reason about ontological knowledge, while in database is trivially sticky since there are no join variables in its body. applications they usually serve as expressive query languages It is a natural question to ask whether the above two in- that go beyond standard first-order queries. A prominent rule- herently different classes of existential rules can be consoli- based formalism is Datalog [Abiteboul et al., 1995]. Even dated into a single formalism. This has been thoroughly in- though Datalog is quite powerful, with a variety of differ- vestigated in [Gottlob et al., 2013]. It has been observed that ent applications, it is widely agreed that its inability to in- the naive combination of guardedness and stickiness leads to fer the existence of new values that are not already in the in- the undecidability of query answering. This led to the notion put database is a crucial limitation [Patel-Schneider and Hor- of tameness, which provides an elegant and flexible way for rocks, 2007]. Existential rules (a.k.a. tuple-generating depen- taming the interaction between guarded and sticky rules. The dencies and Datalog± rules), overcome this limitation by en- essence of tameness is as follows: sticky rules are not allowed riching Datalog with existential quantification in rule-heads. to trigger the guard of a guarded rule. It is easy to verify that However, this leads to the undecidability of the main algo- the above two rules jointly satisfy the tameness condition. rithmic tasks, and, in particular, of conjunctive query (CQ) Conjunctive query answering under the tamed combination answering [Beeri and Vardi, 1981; Cal`ı et al., 2013], i.e., the of guardedness and stickiness is by now well-understood. A problem of checking whether a CQ is entailed by every model sophisticated hybrid query answering algorithm, which relies of an extensional database and a set of existential rules. on a combination of forward and backward reasoning, has This negative outcome has led to a flurry of research ac- been devised in [Gottlob et al., 2013], which led to optimal tivity for identifying restrictions on existential rules that en- complexity results: 2EXPTIME-complete in combined com- sure the decidability of query answering. Two such restric- plexity, and PTIME-complete in data complexity. Notice that tions, which are of central importance for the present work, the analysis performed in [Gottlob et al., 2013] considers un- are guardedness [Baget et al., 2011; Cal`ı et al., 2013] and restricted models. However, in many KR applications the do- stickiness [Cal`ı et al., 2012]. It is well-known that guarded- main of interest is actually finite, and thus it is more realistic 1831 Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence (IJCAI-18) to reason over finite models. This has been already observed Databases and conjunctive queries. An instance is a (pos- in the KR community, and there are several works that focus sibly infinite) set of atoms with constants and nulls, while a on finite model reasoning; see, e.g., [Amendola et al., 2017; database is a finite set of facts. The active domain of an in- Calvanese, 1996; Iba´nez-Garc˜ ´ıa et al., 2014; Rosati, 2008]. stance I, denoted dom(I), is the set of all terms in I. Moreover, for intelligent database applications, the finiteness A( conjunctive query (CQ)) is a formula of the form q(¯x) := of the models is a key assumption. The question that comes 9y¯ R1(¯v1) ^ · · · ^ Rm(¯vm) , where each Ri(¯vi) is an atom up is whether finite query answering, i.e., query answering fo- without nulls, each variable mentioned in the v¯i’s appears ei- cussing on finite models, under tameness remains decidable, ther in x¯ or y¯, and x¯ are the free variables of q. If x¯ is empty, and, if this is the case, what is the exact complexity. Closing then q is a Boolean CQ. As usual, the evaluation of CQs is de- this non-trivial open problem is the main goal of this work. fined in terms of homomorphisms. Let I be an instance, and The main outcome of our analysis is that finite query an- q(¯x) a CQ as above. The evaluation of q(¯x) over I, denoted swering under tameness has the same complexity as (unre- q(I), is defined as the set of tuples c¯ 2 Cjx¯j for which there stricted) query answering. This is obtained by showing that is a homomorphism h such that h(q(¯x)) ⊆ I and h(¯x) =c ¯. the tamed combination of guardedness and stickiness enjoys Notice that, by abuse of notation, we sometimes treat a con- finite controllability, i.e., finite and unrestricted query an- junction of atoms as a set of atoms. AWunion of conjunctive swering coincide. This exploits the fact that both guardedness queries (UCQ) is a formula q(¯x) := 1≤i≤n qi(¯x), where and stickiness enjoy finite controllability [Bar´ any´ et al., 2014; each qi(¯x) is a CQ. The evaluation ofS q(¯x) over I, denoted Gogacz and Marcinkowski, 2017]. At this point, let us q(I), is the set of tuples of constants q (I). say that, in general, combining guardedness with unguarded 1≤i≤n i classes of existential rules that are finitely controllable, by Tuple-generating dependencies. A tuple-generating depen- following the same approach as tameness, does not guaran- dency((TGD) (a.k.a. existential) rule) is a sentence of the form tee finite controllability. As we show later, there exist finitely 8x¯8y¯ ϕ(¯x; y¯) ! 9z¯ (¯x; z¯) , where ϕ, are (non-empty) controllable classes that their tamed combination with guard- conjunctions of atoms without constants and nulls. We write edness leads to classes that are not finitely controllable. Thus, this TGD as ϕ(¯x; y¯) ! 9z¯ (¯x; z¯), and use comma instead to establish the above result, we need to perform a detailed of ^ for joining atoms. We call ϕ and the body and head model-theoretic analysis based on guardedness and sticki- of the TGD, respectively. An instance I satisfies the TGD σ ness. Our results can be summarized as follows: above, written I j= σ, if, whenever there is a homomorphism I We first focus on a subclass of tame rules, obtained by h such that h(ϕ(¯x; y¯)) ⊆ I, then there is h0 that agrees with posing a stratification condition on guarded and sticky rules, h on x¯ such that h0( (¯x; z¯)) ⊆ I. The instance I satisfies a and show that is finitely controllable. set Σ of TGDs, written I j= Σ, if I j= σ for each σ 2 Σ. Let I We then establish that a tame set of guarded and sticky TGD be the class of finite sets of TGDs. Let us clarify that in rules can be rewritten as a stratified one that preserves fi- the rest of the paper we work only with finite sets of TGDs. nite answers, and thus tameness ensures finite controllability. This result immediately implies that finite query answering Query answering under TGDs. For a database D and a set ⊇ under tame guarded and sticky rules is 2EXPTIME-complete Σ of TGDs, a model of D and Σ is an instance I D such j in combined complexity, and PTIME-complete in data com- that I = Σ.